R-L circuit/inductance problem

[bdh2991]

Homework Statement

An RL circuit has a 60V battery, a 42H inductor, a 24Ω resistor and a switch, all in series. Initially the switch is open and there is no magnetic flux in the inductor. At t=0 the switch is closed. When the inductor voltage is 24 V, the time t is closest to?

Homework Equations

i(t) = I(1-e^-t/$\tau$)

$\epsilon$=IR

The Attempt at a Solution

i'm confused by this problem for some reason but what i tried to do was set 60/24 to solve for initial current but i guess if this is a growth problem so I'm not sure what else to do...the answer i got was t=0.9s but i know that's not right

 

[NascentOxygen]

Hi bdh2991! Your equation i(t) = I(1-e^-t/τ ) is spot on!

⚑ What do you calculate to be the value of τ in this circuit? (Incidently, for τ what would be its units?)

⚑ When the voltage across the inductor is 24V, how many volts would there be across the resistor?

⚑ In your equation for i(t), how will you determine what value to use for I?

 

[bdh2991]

$\tau$would = L/R but i believe its units are seconds, numerically it would be 1.75s

I was thinking the voltage across the resistor would be 60= V + 24, so 36V

and the value for I is what I'm confused about, I thought that I is the initial current so it would just be $\epsilon$ = IR, 60=I(24), so I = 2.5?

 

[NascentOxygen]

$\tau$would = L/R but i believe its units are seconds, numerically it would be 1.75s ✔

I was thinking the voltage across the resistor would be 60= V + 24, so 36V ✔

and the value for I is what I'm confused about, I thought that I is the initial current so it would just be $\epsilon$ = IR, 60=I(24), so I = 2.5?

Looking at the equation for i(t), what value of time t would get rid of the exponential term, i.e., make the exponential term = 0?

 

[bdh2991]

at t=0 the exponential would be e^(0) which would equal 1 so i(t) = I (0) = 0

 

[NascentOxygen]

at t=0 the exponential would be e^(0) which would equal 1 so i(t) = I (0) = 0

Correct. The current starts from a very low value. But the question I asked was:

what value of time t would get rid of the exponential term, i.e., make the term e^-t/τ = 0?

 

[NewtonianAlch]

I got 0.83 seconds, but I didn't check it properly.

 

[bdh2991]

i thought and e function could never = 0 ?

why do i feel like i should have gotten what your trying to tell me already

 

[NascentOxygen]

i thought and e function could never = 0 ?

Well, it approaches zero. I'm angling for either 0, τ, or ∞. Which one?

And when e^-t/τ =0, what does your i(t) equation reduce to?

 

[NewtonianAlch]

I'm interested to know now too, I get 1.922s now and I am supposed to remember this stuff!

 

[bdh2991]

i see when t approaches infinity then the equation reduces to i(t) = I, or i guess stabalizes

 

[NewtonianAlch]

i see when t approaches infinity then the equation reduces to i(t) = I, or i guess stabalizes

So what would be the current at t = infinity?

 

[bdh2991]

I = 24/24 so I = 1?

 

[NewtonianAlch]

I = 24/24 so I = 1?

No. What would the circuit look like at t = infinity? I.e. what is the inductor acting like at t = infinity? (Property of an inductor)

 

[NascentOxygen]

i see when t approaches infinity then the equation reduces to i(t) = I, or i guess stabalizes

Yes, most DC circuits given enough time do reach a steady current and voltage.

Doing well.

From your knowledge of DC and inductors in circuits, you know that the voltage across an inductor is di/dt. But when circuit currents and voltages have all settled down to steady values, di/dt = 0 so the voltage across every inductor is 0.

As you can see, I is the final current, the current after infinite time has elapsed.

At t=∞, with the voltage across the inductor equal to zero, all the voltage appears across the resistor. So you can calculate for your circuit's voltage and resistor values that final circuit current, I has a value = https://www.physicsforums.com/images/icons/icon5.gif

 

[bdh2991]

ok so the if the voltage across the conductor is 0 then 60= v +0, therefore v = 60 and R stays the same 24 ohms so I = 2.5 A?

 

[NascentOxygen]

ok so the if the voltage across the conductor is 0 then 60= v +0, therefore v = 60 and R stays the same 24 ohms so I = 2.5 A?

conductor? You mean inductor?

Yes, I=2.5A.

Knowing values for I and τ, you have found the equation exactly describing i(t) from t=0 through to t=∞.

That's all you need to be able to finish the question. The remainder is just mathematical manipulation. Are you okay with it now?

 

[bdh2991]

Yes i think i got it now but just to make certain

To get the actual answer it would be 1=2.5(1-e^-t/1.75) then just solve for t

 

[NascentOxygen]

Yes i think i got it now but just to make certain

To get the actual answer it would be 1=2.5(1-e^-t/1.75) then just solve for t

You haven't explained where that 1 on the LHS magically appeared from.

The equation is i(t) = 2.5(1 – e^-t/1.75) where i(t) is the current through each of L and R. You are asked to determine t when a particular voltage = 24V. You haven't shown an equation relating voltage to time for any element, yet, so you need to come up with one. Write it out explicitly. (Hint: it's easy. )

And there was a further hint back when I asked:

⚑ When the voltage across the inductor is 24V, how many volts would there be across the resistor?

See how you go now. Don't forget, there is always more than one way to solve a circuit problem. If you can see two ways, then you have a means for checking your first solution. :shy:

 

[bdh2991]

i got 1 from taking the voltage across the inductor which would be 24v and divided it by the resistance 24 ohms then substituting it in for the current. if you were to use the voltage of the resister, 36v, then divide by the resistance of 24 ohms you would get i = 2.5 A for that instant but why would you use the voltage across the resistor to solve for the current across the inductor?

i'll never understand physics lol

 

[NascentOxygen]

i got 1 from taking the voltage across the inductor which would be 24v and divided it by the resistance 24 ohms then substituting it in for the current.

Did that sound logical to you?

if you were to use the voltage of the resister, 36v, then divide by the resistance of 24 ohms you would get i = 2.5 A for that instant

Ah! Isn't that precisely what we need? The circuit current at that moment? Anyway you can determine it correctly will do, it's all the same current.

but why would you use the voltage across the resistor to solve for the current across the inductor?

Because those elements share the same current! That's a really good reason.

i'll never understand physics lol

Are you saying my effort has been to no avail?

Could I ask that you do yourself a favour and never refer to "the current across ..." anything. Always it's "current through" an element, and it's "voltage across" an element. If you observe this way of phrasing it, it will serve as a subtle guide to correct analysis, and maybe help you to come to an even better understanding of physics!

You're doing well. Stick with it.

... I have to leave now. Back in 6 hrs ...

 

[NewtonianAlch]

Note, that the current at that instant in the circuit is not 2.5A. That is the final current.

36/24 = 1.5 A

 

[bdh2991]

i think i understand it now...the thing that confused me most was that it said the voltage across the inductor was 24v which made me think to use that to solve for current but now i know not to do that lol

 

[NewtonianAlch]

The standard formula linking current and voltage for an inductor is

V = L(di/dt)

So to get the current across the inductor, you'd have to get di/dt = V/L

And then you'd have to integrate to solve for i. Which you don't want to do for no reason.

Since you have the voltage across the resistor and you know that the current flow in a series circuit is going to be the same, you simply have to get the resistor current, and that's going to be current flowing in the circuit.

 

[NascentOxygen]

i think i understand it now...the thing that confused me most was that it said the voltage across the inductor was 24v which made me think to use that to solve for current but now i know not to do that lol

Still here, I'm delayed a bit.

You certainly can use the voltage across the inductor formula, and determining v_L(t) just involves calculating di/dt which is easy enough for this example here, then equate v_L(t) to 24V and solve for t! No problem at all, and almost as easy as solving for v_R(t)=36 as I led you to do here.

You could try both, and confirm that the answers agree.

NewtonianAlch has misled you, there is no integration involved.

 

[NewtonianAlch]

I do apologise! Yes, I see you can just use di/dt without the need to integrate.

 

[bdh2991]

which V_L equation are you talking about?

i know the kirchhoffs ε - IR - Ldi/dt = 0

but i don't believe that is the same equation your talking about

 

[NascentOxygen]

I was using v_L to denote the voltage across the inductor, L, viz., v_L=L.di/dt

(Using a subscribe to denote the element is a physics convention I should have pointed out.)